A possible
way of analysis of records of seismic and electrical signals is an application
of the classical interpolation theory [13], and also welldeveloped methods of
the theory of information transmission [4,5].
The authors had at their disposal
seismic and electrical records. They represented tables of figures – signals,
recorded by corresponding devices at discrete moments of time. At that, for each
record of a seismic signal (_{} wave) there were two records of an electric signal (a change
of electrical resistivity of environment): one in orientation NS, the other
in orientation EW.
To pass
from a discrete set of tabular data to its analytical representation the
Lagrange interpolation procedure was used [1,2]. This procedure to some extent amplifies
the missing data, allowing to restore a continuous realtime signal according
to an available discrete record to any desired degree of accuracy. It should be
taken into account that any accompanying noise, related to the signal, would be
also recorded. Thus, prior to signals' analysis, it would be desirable to
remove any noise. Below S(x) – is a continuous seismic signal, and
E(x) – a continuous electrical signal. According to [5]
it can be recorded as
 (1) 
Here S_{i} , E_{i} – tabular values of measured seismic and electrical
signals, _{} – a current coordinate (recording
time). The number records in interval _{}, is equal to N + 1. Supposing that out of the interval all S_{i} , E_{i} equal to zero, then series (1) can be replaced by final sums
with summation of series from i = 0 to i = N (assuming that electrical and
seismic signals were recorded simultaneously, with a uniform pitch).
According
to [5], an analytical dependence can be constructed, expressing electric signal
through a seismic one. A corresponding formula (transfer function [5]) can be
represented in the following way
 (2) 
The theory of Lagrange interpolation [1, 5] proves that function E(x) can
be reconstructed by function S(x) with the help of transfer function
(2) to any desired degree of accuracy in the following way
 (3) 
Formula (3) has one obvious
shortcoming – it is rather complicated. It can be simplified, obtaining the following
expression, connecting electrical and seismic signals.
 (4) 
Parameters a and k could be chosen in such a way that
function E^{*}(x) will be very closely approximated to
function E(x). There are different approaches to
defining these parameters. For example, in the present work parameter a was
defined (graphically) according to the condition of a maximum
of the correlation coefficient between initial electrical record and its
approximate representation (4).
The fact that passing of wave P entails a drop in rock electrical
resistivity is well known [6] and can be explained from a physical point of
view. At the moment of passing of a tensile wave microfissures open and are
filled with water (fluid), followed by a drop in electrical resistivity in
proportion to a degree of opening of micropores. Parameter of shift a in
formula (4) in fact represents the time, required for filling micropores with water.
References
1. Levin B. Y. (1956), Distribution of Finite Functions' Roots. M., Nauka, 682 p.
2. Akhiezer N. I. (1965), Lectures on the Theory of Approximations. M., GIFML, 407 p.
3. Kirillov A.A., Gvishiani A. (1982), Theorems and problems in functional
analysis, New YorkHeidelbergBerlin, SpringerVerlag. Seria "Problem books in
mathematics", 347 p.
4. Gvishiani A., Dubois J. (2002), Artificial
Intelligence and Dynamic Systems for Geophysical Applications. SpringerVerlag, Paris, 350 p.
5. Yakovlev Y. I., Khurgin V. P. (1971), Finite Functions in Physics and Technology. M, GIFML, 408 p.
6. G. A. Sobolev, A. V. Ponomarev. Physics of Earthquakes and Precursors. M., Nauka. 2003. 270 p.
